Wednesday, November 17, 2010

Herbert Scarf's 1964 Lectures: An Eyewitness Account

In the fourth volume of The Makers of Modern Economics is a fascinating autobiographical essay by Duncan Foley that traces the arc of his career as an economist and reflects upon developments in the discipline over the past four decades. Duncan describes his first exposure to economics at Swarthmore, his interactions with Tobin as a graduate student at Yale, the introduction in his doctoral dissertation of a concept of equity (now called envy-freeness) that does not depend on interpersonal comparisons of utility, his enormously fruitfulcollaboration with Miguel Sidrauski at MIT on the microfoundations of macroeconomics, his disillusionment with the rational expectations revolution, and his growing interest in heterodox economics at Stanford and subsequently at Barnard and Columbia.

There's enough material there for several interesting posts, but here I'll confine myself to reproducing Duncan's vivid recollection of a two semester course in mathematical economics taught by Herbert Scarf in 1964 (links added):

After the free pursuit of individual learning fostered by the Swarthmore Honors program, I found the return to traditional classroom teaching at Yale a difficult transition... I was frustrated in these courses not just by the tedium and ineffciency of the class lecture style, but by the tendency for instructors who knew a great deal about the substance and practice of their subjects to waste time rehearsing mathematical and theoretical topics they did not understand very well and often misconstrued...

The great exception to this pattern of misdirected pedagogy was Herbert Scarf's year-long course in Mathematical Economics. Scarf knew this material as well as anyone in the world, and had the gifts of patience, clarity of exposition, and personal charisma to convey it brilliantly and effectively. Scarf's teaching was a revelation to me of what could be accomplished in the classroom, with the appropriate attention to systematic organization, consistently careful preparation, and a judicious balance of lecture and discussion to maintain contact with the level of students' understanding. My notes from this course comprise a better and more complete reference for the topics than any book that has since been published.

The passage of time has revealed that the content of Scarf's course was just as remarkable in its depth and insight as the presentation. Remaining mostly within the realm of finite-dimensional spaces, and emphasizing duality and practical algorithms for the construction of solutions, Scarf gave a thorough tutorial on the mathematics of optimization, starting with linear programming via the simplex method and continuing through Kuhn-Tucker theory, dynamic programming, turnpike theory through Roy Radner's algorithmic approach, and integer programming. Since a huge proportion of economic models boil down to an optimization problem, this survey effectively unified and clarified an immense range of economics for the student. When Peter Diamond was working with James Mirrlees on the problem of optimal taxation (Diamond and Mirrlees, 1971a,b), for example, Scarf's approach helped me to grasp the relation between the complexity of their comparative statics results and the nonconvex structure of the constraint set (the intersection of the set of allocations that are resource and technology-feasible and those that can be supported by distorting taxes) in this problem. The study of these formal problems also convinced me that most economic theory depends on strong assumptions of convexity to assure the tractability of the resulting optimization problem, and that in situations where convexity is inherently absent or implausible it is very difficult to make much progress by traditional methods.

Scarf's course continued with a systematic review of general equilibrium theory, starting from the separating hyperplane approach to the Second Welfare Theorem, and including Gérard Debreu's proof (1959) of existence of a competitive equilibrium, the first presentation of Scarf's algorithmic approach to the calculation of competitive equilibria (1973), the theory of the core and its asymptotic equivalence to competitive equilibrium, and Scarf's own crucial counterexamples to the stability of competitive equilibrium under tâtonnement dynamics with more than two commodities (1960). The critical lesson Scarf emphasized in this discussion was the fact that the competitive equilibrium cannot, except in special cases such as representative agent economies, be represented as the solution of a mathematical programming problem. In other words, the Walrasian system does not generally admit a potential function. As a corollary to this observation we see that the comparative statics of competitive general equilibrium theory inherently lacks the organizing structure of convex programming, so that, for example, equilibrium prices are not in general monotonic functions of endowments. These observations planted the seeds in my mind of what grew to be grave doubts about the Walrasian system. These doubts do not focus on the logical consistency of the system, but on its adequacy as a useful representation of real economic relations...

In retrospect we can see that Scarf's course mapped out the whole development of high economic theory for the next twenty or twenty-five years. The theoretical literature of this period has largely been concerned with generalizing the concepts he taught to more sophisticated commodity spaces (such as infinite-dimensional spaces and spaces of stochastic processes), and rediscovering the general properties and limitations of competitive equilibrium theory in these contexts. This has been a source of both wonder and concern to me. I am amazed at how prescient a mind like Scarf's can be about the future development of a field, guided purely by superb mathematical instincts. But what does this imply about the theoretical fertility of economics during this period? If the core theoretical ideas that have dominated the field since were all present in the Yale classroom in 1964, it suggests that economic theory has been in a scholastic, formalistic phase of development during this period, primarily focusing on working out increasingly esoteric implications of well-established concepts.

Duncan tells me that he still has his notes from this course and that Scarf, who recently retired from teaching, remains full of vigor.

In subsequent posts I hope to discuss Duncan's reflections on the microfoundations of macroeconomics, his work with Sidrauski, his concern that the rational expectations revolution was a step backwards in the development of the theory, and his view that "some break with the full Walrasian system along temporary equilibrium lines is necessary as a foundation for a distinct macroeconomics." (The Hicksian concept of temporary equilibrium allows for asset market clearing in the face of heterogeneous beliefs and mutually inconsistent intertemporal plans.) These are themes that I have touched upon in previousposts and would like to revisit soon. In the meantime, let me repeat my plea to the fellows of the Economteric Society to nominate Duncan for election to their ranks.

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Update (11/18). Glenn Loury writes in to say:

I never had much interaction with Scarf, but his pedagogic virtuosity and mastery of mathematical economics circa 1970 reminds me of... Stanley Reiter, whom I encountered as a raw assistant professor at Northwestern in the 1970s. Stan, a close friend and occasional collaborator with Leo Hurwicz, was director of the Math Center at Northwestern (forerunner of MEDS), and in the late 1970s had a huge impact on young scholars like Paul Milgrom, Bengt Holmstrom, Mark Satterthwaite and Roger Myerson...

I don't think I agree with the claim that much of "high economic theory" since the 60s has been dotting "i's" and crossing "t's". That was true through the mid-seventies, perhaps, but the asymmetric information, mechanism design, incomplete contract theory revolutions (Hurwicz/Myerson/Maskin, eg.) -- and the emergence of deeply insightful applied theory in a variety of fields from labor and I/O to money, finance and trade suggest otherwise to me.

I basically agree with Glenn on this latter point but, in Duncan's defense, the focus of his essay was on the microfoundations of macroeconomics and the futility of simply aggregating the Walrasian system. And on this dimension I think that progress has been limited at best.

I too sat in Herb Scarf's Yale Micro Theory classroom and still remember the stunned awe that I and my classmates felt at the end of his first lecture with us, which happened to be on the simplex algorithm.

My only regret is that that semester at Yale (1990) we only got a handful of micro lectures from Scarf and so did not get the full "systematic review of general equilibrium theory" that Foley mentions.

I have little to say to improve on Duncan's glowing description of a Scarf lecture except to note that by 1990 the Hillhouse basement classroom had smooth sliding blackboards (which I do not imagine they had in 1964). This meant that there were always three blackboards in use, as he could fill one blackboard full of equations and slide it to conceal or reveal what had been written before. One of the things I recall most vividly is how artfully and efficiently Scarf used those boards, and how rarely he used the eraser. A lecture which might have started with definitions and theory that might have taken a detour through an expertly chosen example to reinforce intuition would in the end always return, with the smoothest glide of a hand to reveal again exactly the right portion of the board to bring the lecture full circle back to the climactic point he wanted. Everything seemed expertly choreographed and timed down to the very last second.

I hope that other former students of Scarf will somehow stumble upon this post.

4 comments:

I too sat in Herb Scarf's Yale Micro Theory classroom and still remember the stunned awe that I and my classmates felt at the end of his first lecture with us, which happened to be on the simplex algorithm. My only regret is that that semester at Yale (1990) we only got a handful of micro lectures from Scarf and so did not get the full "systematic review of general equilibrium theory" that Foley mentions. I have little to say to improve on Duncan's glowing description of a Scarf lecture except to note that by 1990 the Hillhouse basement classroom had smooth sliding blackboards (which I do not imagine they had in 1964). This meant that there were always three blackboards in use, as he could fill one blackboard full of equations and slide it to conceal or reveal what had been written before. One of the things I recall most vividly is how artfully and efficiently Scarf used those boards, and how rarely he used the eraser. A lecture which might have started with definitions and theory that might have taken a detour through an expertly chosen example to reinforce intuition would in the end always return, with the smoothest glide of a hand to reveal again exactly the right portion of the board to bring the lecture full circle back to the climactic point he wanted. Everything seemed expertly choreographed and timed down to the very last second.

Rajiv - long time! Just when I thought you'd abandoned the blogosphere.

I look forward to your posts on Duncan Foley's work as I'm not familiar of many of his papers you've linked to. But a while back, I read Peter Albin's excellent book "Barriers and Bounds to Rationality" which Duncan Foley edited and contributed the introduction to. In that introduction, he touches upon some of the topics you mention (Walrasian system and representative agents, robustness of rational choice results). The book itself is also one of my favourites although it may be a bit too heterodox for some tastes.